Group #38742824 — subgroups

Subgroups — #38742824

Order 25216.

236 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27161557	C₂	1	1	yes	maximal, characteristic
2	C₂×#20263245	C₂	1	1	yes	maximal, characteristic
3	#27160851	C₂	1	2	yes	maximal
4	#27160850	C₂	1	2	yes	maximal
5	C₂×#101×C₁₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#15691342	C₂²	1	2	yes	—
7	Q₈×Dic₁₉₇	—	2	2	no	—
8	Q₈×Dic₁₉₇	—	2	2	no	—
9	#20263285	C₄	1	4	yes	—
10	#20263285	C₂²	1	1	yes	characteristic
11	#20263285	C₂²	1	1	yes	characteristic
12	#20263245	—	2	2	no	—
13	C₂²×Q₈×C₁₉₇	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₁₉₇	C₂²	1	2	yes	—
15	#101×C₁₉₇	—	2	2	no	—
16	#15691342	—	2	2	no	—
17	#15691342	—	2	2	no	—
18	C₂×#13096802	—	2	2	no	—
19	C₄⋊Dic₁₉₇	—	2	4	no	—
20	C₄⋊Dic₁₉₇	—	2	1	no	—
21	C₄⋊Dic₁₉₇	—	2	1	no	—
22	C₂²⋊Dic₁₉₇	C₂×C₄	1	2	yes	—
23	C₂²⋊Dic₁₉₇	C₂×C₄	1	2	yes	—
24	C₄×Dic₁₉₇	—	2	4	no	—
25	C₄×Dic₁₉₇	—	2	1	no	—
26	C₄×Dic₁₉₇	—	2	1	no	—
27	C₂×Q₈×C₁₉₇	—	2	2	no	—
28	C₂×Q₈×C₁₉₇	—	2	2	no	—
29	C₂×Q₈×C₁₉₇	D₄	1	2	yes	—
30	C₂×Q₈×C₁₉₇	D₄	1	2	yes	—
31	C₂²×C₄×C₁₉₇	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
33	M₁₆×C₁₉₇	—	2	2	no	—
34	M₁₆×C₁₉₇	—	2	2	no	—
35	C₂×C₈×C₁₉₇	—	2	2	no	—
36	C₂×Dic₁₉₇	—	2	2	no	—
37	C₂×Dic₁₉₇	—	2	2	no	—
38	#13096802	—	4	2	no	—
39	C₂³×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
40	Q₈×C₁₉₇	—	2	4	no	—
41	Q₈×C₁₉₇	—	2	4	no	—
42	C₂×C₄×C₁₉₇	—	2	2	no	—
43	C₂×C₄×C₁₉₇	—	2	2	no	—
44	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	4	yes	—
45	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₉₇	#41	1	2	yes	—
48	C₂×C₄×C₁₉₇	#41	1	2	yes	—
49	C₈×C₁₉₇	—	4	2	no	—
50	C₂²×C₁₉₇	—	4	1	no	—
51	C₂²×C₁₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₁₉₇	#117	1	1	yes	characteristic
53	C₂²×C₁₉₇	#117	1	1	yes	characteristic
54	C₄×C₁₉₇	—	2	4	no	—
55	C₄×C₁₉₇	#123	1	2	yes	—
56	C₄×C₁₉₇	#123	1	2	yes	—
57	Dic₁₉₇	—	4	2	no	—
58	Dic₁₉₇	—	4	2	no	—
59	C₂×C₁₉₇	—	2	1	no	—
60	C₂×C₁₉₇	—	2	1	no	—
61	C₂×C₁₉₇	—	2	1	no	—
62	C₂×C₁₉₇	#381	1	1	yes	characteristic
63	C₁₉₇	#1830	1	1	yes	characteristic, complemented, Sylow
64	#1830	—	197	1	no	maximal, Sylow
65	#519	—	197	1	no	—
66	C₂×#101	—	197	1	no	—
67	C₂×#101	D₁₉₇	1	1	yes	characteristic
68	#344	—	197	2	no	—
69	#344	—	197	2	no	—
70	C₂²×Q₈	C₂×D₁₉₇	1	1	yes	characteristic
71	C₂×M₁₆	—	197	2	no	—
72	C₂×M₁₆	C₂×D₁₉₇	1	2	yes	—
73	C₄×Q₈	—	394	2	no	—
74	C₄×Q₈	—	394	2	no	—
75	#117	—	197	4	no	—
76	#117	—	197	1	no	—
77	#117	—	197	1	no	—
78	#101	—	2	2	no	—
79	#101	—	394	2	no	—
80	C₂×Q₈	—	2	2	no	—
81	C₂×Q₈	—	2	2	no	—
82	C₂×Q₈	C₂²⋊D₁₉₇	1	2	yes	—
83	C₂×Q₈	D₇₈₈	1	2	yes	—
84	C₂²×C₄	C₄×D₁₉₇	1	2	yes	—
85	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
86	M₁₆	—	2	2	no	—
87	M₁₆	—	2	2	no	—
88	M₁₆	—	394	2	no	—
89	M₁₆	—	394	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	394	2	no	—
92	C₄⋊C₄	—	394	4	no	—
93	C₄⋊C₄	—	394	1	no	—
94	C₄⋊C₄	—	394	1	no	—
95	C₂²⋊C₄	—	197	2	no	—
96	C₂²⋊C₄	—	197	2	no	—
97	C₄²	—	394	4	no	—
98	C₄²	—	394	1	no	—
99	C₄²	—	394	1	no	—
100	C₂³	C₂×C₄×D₁₉₇	1	1	yes	characteristic
101	Q₈	—	2	4	no	—
102	Q₈	—	2	4	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	394	2	no	—
106	C₂×C₄	—	394	2	no	—
107	C₂×C₄	#15691335	1	4	yes	—
108	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
109	C₂×C₄	C₂×D₇₈₈	1	1	yes	characteristic
110	C₂×C₄	#15691357	1	2	yes	—
111	C₂×C₄	#15691357	1	2	yes	—
112	C₈	—	4	2	no	—
113	C₈	—	788	2	no	—
114	C₂²	—	4	1	no	—
115	C₂²	C₂×#15691335	1	1	yes	characteristic
116	C₂²	#20263282	1	1	yes	characteristic
117	C₂²	#20263282	1	1	yes	characteristic
118	C₄	—	2	4	no	—
119	C₄	—	788	2	no	—
120	C₄	—	788	2	no	—
121	C₄	#20263307	1	2	yes	—
122	C₄	#20263310	1	2	yes	—
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	—	2	1	no	—
126	C₂	#27160940	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27161557

C₂

yes

maximal, characteristic

C₂×#20263245

C₂

yes

maximal, characteristic

#27160851

C₂

yes

maximal

#27160850

C₂

yes

maximal

C₂×#101×C₁₉₇

C₂

yes

maximal, characteristic

C₂×#15691342

C₂²

yes

—

Q₈×Dic₁₉₇

—

Q₈×Dic₁₉₇

—

#20263285

C₄

yes

—

#20263285

C₂²

yes

characteristic

#20263285

C₂²

yes

characteristic

#20263245

—

C₂²×Q₈×C₁₉₇

C₂²

yes

characteristic

C₂×M₁₆×C₁₉₇

C₂²

yes

—

#101×C₁₉₇

—

#15691342

—

#15691342

—

C₂×#13096802

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

—

C₂²⋊Dic₁₉₇

C₂×C₄

yes

—

C₂²⋊Dic₁₉₇

C₂×C₄

yes

—

C₄×Dic₁₉₇

—

C₄×Dic₁₉₇

—

C₄×Dic₁₉₇

—

C₂×Q₈×C₁₉₇

—

C₂×Q₈×C₁₉₇

—

C₂×Q₈×C₁₉₇

D₄

yes

—

C₂×Q₈×C₁₉₇

D₄

yes

—

C₂²×C₄×C₁₉₇

C₂×C₄

yes

—

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

M₁₆×C₁₉₇

—

M₁₆×C₁₉₇

—

C₂×C₈×C₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

#13096802

—

C₂³×C₁₉₇

C₂²×C₄

yes

characteristic

Q₈×C₁₉₇

—

Q₈×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

#41

yes

—

C₂×C₄×C₁₉₇

#41

yes

—

C₈×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

C₂×C₂²⋊C₄

yes

C₂²×C₁₉₇

yes

C₂²×C₁₉₇

yes

C₄×C₁₉₇

—

C₄×C₁₉₇

#123

yes

—

C₄×C₁₉₇

#123

yes

—

Dic₁₉₇

—

Dic₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

#381

yes

characteristic

C₁₉₇

#1830

yes

characteristic, complemented, Sylow

#1830

—

197

maximal, Sylow

#519

—

197

—

C₂×#101

—

197

—

C₂×#101

D₁₉₇

yes

characteristic

#344

—

197

—

#344

—

197

—

C₂²×Q₈

C₂×D₁₉₇

yes

characteristic

C₂×M₁₆

—

197

—

C₂×M₁₆

C₂×D₁₉₇

yes

—

C₄×Q₈

—

394

—

C₄×Q₈

—

394

—

#117

—

197

—

#117

—

197

—

#117

—

197

—

#101

—

#101

—

394

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₁₉₇

yes

—

C₂×Q₈

D₇₈₈

yes

—

C₂²×C₄

C₄×D₁₉₇

yes

—

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

394

—

M₁₆

—

394

—

C₂×C₈

—

C₂×C₈

—

394

—

C₄⋊C₄

—

394

—

C₄⋊C₄

—

394

—

C₄⋊C₄

—

394

—

C₂²⋊C₄

—

197

—

C₂²⋊C₄

—

197

—

C₄²

—

394

—

C₄²

—

394

—

C₄²

—

394

—

100

C₂³

C₂×C₄×D₁₉₇

yes

characteristic

101

Q₈

—

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

394

—

106

C₂×C₄

—

394

—

107

C₂×C₄

#15691335

yes

—

108

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

characteristic

109

C₂×C₄

C₂×D₇₈₈

yes

characteristic

110

C₂×C₄

#15691357

yes

—

111

C₂×C₄

#15691357

yes

—

112

C₈

—

113

C₈

—

788

—

114

C₂²

—

115

C₂²

C₂×#15691335

yes

characteristic

116

C₂²

#20263282

yes

characteristic

117

C₂²

#20263282

yes

characteristic

118

C₄

—

119

C₄

—

788

—

120

C₄

—

788

—

121

C₄

#20263307

yes

—

122

C₄

#20263310

yes

—

123

C₂

—

124

C₂

—

125

C₂

—

126

C₂

#27160940

yes

characteristic

	subgroup	G/H
center	C₂	#27160940
commutator, prime radical	C₂³×C₁₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₇
Fitting	C₂×#101×C₁₉₇	C₂
socle	C₂×C₁₉₇	#381
cosocle	C₂²×C₄×C₁₉₇	C₂³
hypercenter	C₂×#101	D₁₉₇
hypocenter	C₁₉₇	#1830
LCS	G ▷ C₂³×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#101

subgroup

G/H

center

C₂

#27160940

commutator, prime radical

C₂³×C₁₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₇

Fitting

C₂×#101×C₁₉₇

C₂

socle

C₂×C₁₉₇

#381

cosocle

C₂²×C₄×C₁₉₇

C₂³

D₁₉₇

C₁₉₇

G ▷ C₂³×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#101

N	H ≅ G/N	type
63. C₁₉₇	#1830	semidirect

H ≅ G/N

type

63. C₁₉₇

#1830

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#27161557	2
C₂×#20263245	2
#27160851	2
#27160850	2
C₂×#101×C₁₉₇	2
#1830	197

subgroup

index

C₂×#101×C₁₉₇

197

Subgroups — #38742824

Conjugacy classes of subgroups

Complements

Maximal subgroups